Question

If \[ |\vec a|=2k, \qquad |\vec b|=k \] and \[ |\vec a-\vec b|^2=20k^2-|2\vec a+\vec b|^2, \] then \[ |\vec a\times\vec b| = \ ? \]

• A. \(\sqrt3\,k^2\)
• B. \(k^2\)
• C. \(2\sqrt3\,k^2\)
• D. \(2k^2\)

Hint:

Whenever expressions such as \[ |\vec a+\vec b|^2 \quad\text{or}\quad |\vec a-\vec b|^2 \] appear, expand them using dot products first. The angle between vectors can then be obtained easily.

Answer 1

The Correct Option is A

Concept: The magnitude of the cross product is \[ |\vec a\times\vec b| = |\vec a||\vec b|\sin\theta. \] To determine \(\sin\theta\), we first find the angle between the vectors using the given vector identity.

Step 1: Expand \( |\vec a-\vec b|^2 \) Using \[ |\vec a-\vec b|^2 = |\vec a|^2+|\vec b|^2-2\vec a\cdot\vec b, \] we get \[ = (2k)^2+k^2-2\vec a\cdot\vec b. \] \[ = 5k^2-2\vec a\cdot\vec b. \]

Step 2: Expand \( |2\vec a+\vec b|^2 \) \[ |2\vec a+\vec b|^2 = 4|\vec a|^2+|\vec b|^2+4\vec a\cdot\vec b. \] \[ = 16k^2+k^2+4\vec a\cdot\vec b. \] \[ = 17k^2+4\vec a\cdot\vec b. \]

Step 3: Substitute into the given condition Given, \[ 5k^2-2\vec a\cdot\vec b = 20k^2-(17k^2+4\vec a\cdot\vec b). \] \[ 5k^2-2\vec a\cdot\vec b = 3k^2-4\vec a\cdot\vec b. \] \[ 2k^2 = -2\vec a\cdot\vec b. \] \[ \vec a\cdot\vec b = -k^2. \]

Step 4: Find \(\cos\theta\) \[ \vec a\cdot\vec b = |\vec a||\vec b|\cos\theta. \] \[ -k^2 = (2k)(k)\cos\theta. \] \[ \cos\theta = -\frac12. \] Therefore, \[ \sin\theta = \frac{\sqrt3}{2}. \]

Step 5: Calculate the cross product magnitude \[ |\vec a\times\vec b| = |\vec a||\vec b|\sin\theta. \] \[ = (2k)(k)\left(\frac{\sqrt3}{2}\right). \] \[ = \sqrt3\,k^2. \] Hence, \[ \boxed{\sqrt3\,k^2}. \]